Circle Calculator, Explained

Give one measurement of a circle and read off every other one: radius, diameter, circumference, area, plus the arc, chord, sector and segment for any central angle.

What the circle calculator does

A circle is fixed by a single number. Tell the calculator the radius, and everything else follows: the diameter is twice the radius, the circumference and area come from \pi. The same is true in reverse. If you know the area, you can recover the radius, and from there the rest.

Here is the hook. Suppose a circular table has an area of 2 square meters. What radius board do you need to cut? Rearranging the area formula gives r = \sqrt{2/\pi} \approx 0.7979 meters, so a diameter of about 1.596 meters. One measurement in, four measurements out.

Add an optional central angle and the calculator also returns the arc length, the straight chord across that angle, the pie-slice sector area and the segment area cut off by the chord. Those four are the parts most people get wrong by hand, so they are worth understanding carefully.

When to use it, and when not

Use this calculator whenever a shape is genuinely a circle or a piece of one: a wheel, a pipe cross section, a fillet radius, a pizza slice, a running track curve, a lens aperture. Any single known value gives you the rest.

Do not use it for shapes that only look round. An ellipse has two different radii and its circumference has no simple closed form. A rounded rectangle is straight edges plus corner arcs, so you handle the arcs here and the straight parts separately. For polygons, reach for the Polygon Area Calculator or the Regular Polygon Calculator instead.

Circumference and area are not interchangeable, even though both grow with size. Circumference scales with the radius, area scales with the radius squared. Doubling the radius doubles the circumference but quadruples the area.

The formulas and why they hold

Start with the two anchor formulas. For a circle of radius r:

C = 2\pi r \qquad A = \pi r^2

Here C is the circumference (the distance once around) and A is the area (the region inside). The diameter is d = 2r. The constant \pi \approx 3.14159 is the ratio of circumference to diameter, the same for every circle.

To go the other way, solve for r. From C = 2\pi r you get r = C/(2\pi). From A = \pi r^2 you get r = \sqrt{A/\pi}. The calculator does exactly this internally: convert whatever you gave it into a radius, then compute the rest.

Now the angle-dependent parts. Let \theta be the central angle in degrees. The arc is the fraction \theta/360 of the full circumference:

L = 2\pi r \cdot \frac{\theta}{360}

The sector (the pie slice) is the same fraction of the area:

A_{\text{sector}} = \pi r^2 \cdot \frac{\theta}{360}

The chord is the straight line joining the two endpoints of the arc. Drop a line from the center to the midpoint of the chord and you get two right triangles, each with angle \theta/2 at the center:

c = 2r \sin\!\left(\frac{\theta}{2}\right)

The segment is the sliver between the chord and the arc. Take the sector and subtract the triangle formed by the two radii and the chord:

A_{\text{segment}} = \frac{1}{2} r^2 \left(\frac{\pi \theta}{180} - \sin\theta\right)

The term \pi\theta/180 converts \theta to radians, because the clean form of the segment area uses radians. If trig ratios like \sin are unfamiliar, the Trigonometry Calculator shows how they behave.

Worked example with the demo data

Radius 5, central angle 60 degrees

These are the demo button values: known = radius, value = 5, angle = 60.

Diameter: d = 2 \times 5 = 10.
Circumference: C = 2\pi \times 5 \approx 31.42.
Area: A = \pi \times 5^2 \approx 78.54.
Arc length at 60 degrees: L = 31.42 \times 60/360 \approx 5.236.
Sector area: A_{\text{sector}} = 78.54 \times 60/360 \approx 13.09.
Chord: c = 2 \times 5 \times \sin(30^\circ) = 10 \times 0.5 = 5.
Segment area: \tfrac{1}{2} \times 25 \times (\pi \times 60/180 - \sin 60^\circ) = 12.5 \times (1.0472 - 0.8660) \approx 2.265.

Notice the chord equals the radius here, exactly 5. That is a feature of the 60 degree angle: the two radii and the chord form an equilateral triangle, so all three sides are 5. That triangle relationship is the same one solved by the Triangle Solver.

How arc, chord and segment change with the angle

Fix the radius at 5 and sweep the central angle. The arc grows in a straight line with the angle, since it is just a fraction of a fixed circumference. The chord grows too, but it bends over: it peaks at the diameter of 10 when the angle reaches 180 degrees, then shrinks back. The table shows the pattern.

Radius 5: arc, chord and segment area versus central angle
Angle (deg)	Arc length	Chord	Sector area	Segment area
30	2.618	2.588	6.545	0.2835
60	5.236	5.000	13.09	2.265
90	7.854	7.071	19.63	7.135
120	10.47	8.660	26.18	15.35
180	15.71	10.00	39.27	39.27

At 180 degrees the chord is the full diameter and the segment equals the sector, because the segment is now the whole half circle. That is the check to remember: at 180 degrees, segment area and sector area are identical.

Arc length rises in a straight line while the chord flattens toward the diameter of 10.

Reading and interpreting the results

Every length shares the unit you typed in, and every area is that unit squared. Enter a radius in centimeters and the circumference comes out in centimeters, the area in square centimeters. The calculator does not attach units, so keep track of them yourself.

Sanity-check the relationships. The circumference should be a little more than three diameters, since C/d = \pi \approx 3.14. The area should equal roughly 0.785 of the square that boxes the circle, because \pi/4 \approx 0.7854. For radius 5, that box is 10 \times 10 = 100, and indeed the area is 78.54.

Arc: The curved piece of the circumference spanned by the angle.
Chord: The straight segment joining the two endpoints of that arc.
Sector: The pie slice bounded by two radii and the arc.
Segment: The region between the chord and the arc, smaller than the sector.

Common mistakes

The first trap is mixing radius and diameter. If a spec sheet lists a 10 cm pipe, that is almost always the diameter, so the radius is 5. Feeding 10 as the radius quadruples the area you expect.

The second trap is degrees versus radians. The arc and sector formulas above use degrees through the \theta/360 fraction, but the compact segment formula needs radians. The calculator handles the conversion; if you compute by hand, convert with \text{radians} = \theta \times \pi/180.

Confusing the sector and the segment is common. The sector includes the triangle from the center; the segment does not. For radius 5 at 60 degrees the sector is 13.09 but the segment is only 2.265, because the triangle you subtract has area 10.83.

The third trap is forgetting to square the unit for area, then comparing a length to an area. A circumference of 31.42 cm and an area of 78.54 cm² describe different quantities and should never be added or ranked against each other.

Related calculators

Once you leave a single circle, other tools fit better. For the volume and surface area of a sphere, cone or cylinder, use the 3D Solid Geometry Calculator. When your problem reduces to a right triangle, as the chord construction does, the Right Triangle Calculator handles it directly. To find a circle through given points or work with its equation, try the Coordinate Geometry Toolkit.

Frequently asked questions

How do I find the radius from the area?

Divide the area by \pi and take the square root: r = \sqrt{A/\pi}. For an area of 78.54, that is \sqrt{78.54/3.14159} = \sqrt{25} = 5.

What is the difference between a sector and a segment?

A sector is the full pie slice bounded by two radii and the arc. A segment is only the part between the chord and the arc. The segment is always the smaller of the two for angles under 180 degrees. At 180 degrees they are equal, since both become the half circle.

Why does the chord equal the radius at 60 degrees?

The two radii and the chord form a triangle with a 60 degree apex and two equal sides. That forces the other two angles to 60 degrees as well, making the triangle equilateral, so the chord matches the radius. For radius 5 the chord is exactly 5.

Can I enter an angle greater than 180 degrees?

Yes. The arc and sector formulas keep working up to 360 degrees, where the arc equals the full circumference and the sector equals the whole area. Above 180 degrees the segment becomes the larger piece of the circle.

Does the calculator use units?

No. It works with pure numbers. Whatever unit you enter for the known value carries through to every length, and areas come out in that unit squared.

Try it yourself in the tool